### How well does Transition Mathematics address the content in the selected benchmarks?

This In Brief chart provides profiles showing how this textbook scored on content and instructional quality. For the content profile, the coverage of each specific mathematical idea in the selected benchmark was rated on a 0 to 3 scale (no coverage to substantive coverage). These ratings were then averaged to obtain an overall rating for each benchmark (Most content 2.6-3.0, Partial content 1.6-2.5, Minimal content 0-1.5). For the instruction profile, the score for each instructional category was computed by averaging the criterion ratings for the category. This was repeated for each benchmark, to produce ratings of instructional quality on a 0 to 3 scale (High potential for learning to take place 2.6-3.0, Some potential for learning to take place 1.6-2.5, Little potential for learning to take place 0.1-1.5, Not present 0).

### Transition Mathematics in Brief

 Benchmarks Number Concepts Number Skills Geometry Concepts Geometry Skills Algebra Graph Concepts Algebra Equation Concepts Content      Instructional Categories Identifying a Sense of Purpose      Building on Student Ideas about Mathematics      Engaging Students in Mathematics      Developing Mathematical Ideas      Promoting Student Thinking about Mathematics      Assessing Student Progress in Mathematics      Enhancing the Mathematics Learning Environment      Content Scale Instructional Categories Scale Most content Partial content Minimal content High potential for learning to take place Some potential for learning to take place Little potential for learning to take place Not present

The content ratings are estimates of what the textbook series attempts to present on only these benchmarks and are not an indication of overall content coverage or accuracy. The ratings also do not indicate whether or not the content will be learned. The instructional analysis provides information on the potential the series has for helping students actually learn the concepts and skills it attempts to present.

The following indicates how well Transition Mathematics attempts to address the substance, breadth, and sophistication of the ideas contained in each of the six mathematics benchmarks that were selected for the analysis.

Number Concepts — Most Content The expression a/b can mean different things: a parts of size 1/b each, a divided by b, or a compared to b. (Chapter 9A, grades 6-8, benchmark 5, pg. 213.)

All the benchmark ideas are addressed in this material, although each idea is contained in only one or two lessons. In a lesson on multiplication of fractions, the algebraic definition of division is provided as a(1/b) = a/b with examples and an explanation. Multiplication of fractions is illustrated with diagrams showing "a parts of size 1/b" as areas of rectangles. In the definition of integer division, "a divided by b" is related to real number division by showing that there is a remainder for division. The last part of the benchmark is addressed in a lesson on the ratio comparison model for division, illustrating that a/b means "a compared to b," which is a direct match with the benchmark idea.

Number Skills — Most Content Use, interpret, and compare numbers in several equivalent forms such as integers, fractions, decimals, and percents. (Chapter 12B, grades 6-8, benchmark 2, pg. 291.)

All parts of this benchmark are addressed in many lessons and contexts throughout the book. Early lessons on converting fractions to decimals are extended to include writing mixed numbers as decimals. Later lessons present the procedure of simplifying fractions, which addresses the idea of equivalent fractions. The book presents equivalent ways of writing numbers, such as using powers of ten as exponents and in word form (thousand, billion) and writing expressions with exponents and bases. Significant coverage is devoted to the meaning of percent and the relation among percents, fractions, and decimals. Finally, algebraic approaches are used to convert decimals to fractions and to illustrate slopes as equivalent fractions.

Geometry Concepts — Partial Content Some shapes have special properties: Triangular shapes tend to make structures rigid, and round shapes give the least possible boundary for a given amount of interior area. Shapes can match exactly or have the same shape in different sizes. (Chapter 9C, grades 6-8, benchmark 1, pg. 224.)

Several chapters address the usual properties of triangles, rectangles, trapezoids, parallelograms, and circles. Special properties, such as the rigid structure of triangles and the area relationship to the boundary of a circle are not addressed specifically. The area of a triangle is developed from a rectangle, and polygons are divided into triangles to find their areas. A series of lessons looks at size changes by treating expansions and contractions of line segments, triangles, and polygons using coordinates. The ideas are extended and applied to maps and drawings.

Geometry Skills — Most Content Calculate the circumferences and areas of rectangles, triangles, and circles, and the volumes of rectangular solids. (Chapter 12B, grades 6-8, benchmark 3, pg. 291.)

All ideas of this benchmark are addressed in this material. Formulas are presented for the area of a square and rectangle, the volume of a cube, and the relation of volume to capacity. A more complete treatment is then given for the area model for multiplication, leading to the formula for the area of a rectangle, the area of a right triangle, and the calculation of other areas by decomposing figures into rectangles and triangles. The formulas for circumference and area of a circle are developed, practiced, and applied. Volumes of rectangular solids are developed, and the idea of measuring the size of two-dimensional objects is used to distinguish between calculating area and perimeter, which is then followed by the formula and calculation of areas of triangles, trapezoids, and parallelograms.

Algebra Graph Concepts — Partial Content Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease indefinitely, increase or decrease in steps, or do something different from any of these. (Chapter 9B, grades 6-8, benchmark 3, pg. 219.)

Several parts of this benchmark, such as the variable getting closer to some limiting value and reaching an intermediate minimum or maximum are never addressed. Most of the work on this benchmark is in two chapters; one that deals with data displays and another that presents coordinate graphs and equations. The latter chapter, which is the last chapter in the book, is more advanced and not intended for average students. Examples show decreases or increases over time, and students are asked to interpret the graphs. Several types of graphs illustrate the display of data, but only a few examples address directly the idea of steady or changing rate of increase or decrease. The last chapter briefly addresses the rate of increase, but there is little direct discussion of rates of change beyond noting the differing shapes of the graphs. One lesson deals with step graphs, addressing the part of the benchmark about increases or decreases in steps.

Algebra Equation Concepts — Most Content Symbolic equations can be used to summarize how the quantity of something changes over time or in response to other changes. (Chapter 11C, grades 6-8, benchmark 4, pg. 274.)

In Transition Mathematics, the primary emphasis on symbolic equations is on methods of solution. The main idea of the benchmark, how quantities change, is treated implicitly through solving the equations. Equations of the form x +a =b and a lesson on spreadsheets show changes in quantities. Later, solution methods for other types of equations are addressed, providing examples that show implicitly how quantities change. Linear equations and graphs illustrate rates of change and the effect on the slopes and other characteristics of linear graphs and relationships.